Current sensorless position-tracking control with angular acceleration error observers for hybrid-type stepping motors

This paper exhibits an advanced observer-based position-tracking controller for hybrid-type stepping motors with consideration of parameter and load uncertainties. As the main contribution, a current sensorless observer-based pole-zero cancellation speed controller is devised for the outer loop position-tracking controller including the convergence rate boosting mechanism. The features of this study are summarized as follows; first, the pole-zero cancellation angular acceleration error observer for the inner loop speed controller, second, the pole-zero cancellation speed control forcing the order of the controlled speed error dynamics to be 1, and, third, the outer loop position control incorporating the first-order target tracking system with its convergence rate booster. The resultant effectiveness is verified on a 10-W stepping motor control system.

The major advantages of stepping motors are the elimination of brushes and the use of a simple position regulation method to count the pulse numbers. These allow various industrial position control applications, such as computerized numerical control (CNC) machines, nuclear reactor control rods, robot arms, and printers [1][2][3][4][5][6][7] .
Implementing position and speed regulation with stepping motors is possible without any feedback sensors by counting the pulse numbers and adjusting the pulse frequency; however, their precision is predominantly reliant on the teeth numbers. At high-speeds, a stepping motor can experience mechanical problems such as step-out, resonance, and reversal of speed 8 . To overcome these, a micro-stepping technique with a partial closedloop structure was proposed that determines the voltage update law statically while assuming the current-loop transfer function as 1. The corresponding closed-loop control precision and performance are dependent on the current controller. A proportional-integral-derivative (PID) control constitutes the current-loop for each phase with a well-tuned feedback gain using Bode and Nyquist techniques [8][9][10][11][12] . To maintain the desired performance across a large operation range, the resultant feedback gain must be magnified by increasing the motor speed, which is proportional to the back-electromotive force (EMF) disturbance. Parameter-dependent feed-forward compensators deal with this problem by canceling the motor-speed-dependent disturbance, which can achieve significant performance improvement in the high-speed mode 13 . A novel current-control technique was proposed based on the incorporation of a disturbance observer (DOB) in the sliding-mode control (SMC) to improve the feed-forward terms by reducing the parameter dependence; the proof of closed-loop convergence was presented by the Lyapunov stability theorem 14 . Another recent study established the elimination of current feedback sensors by combining feedback-linearization (FL) control and a passive observer driven by the position error, which included closed-loop stability analysis 15 . The level to which parameters depend on these techniques can be lowered by using the novel online parameter identifiers as in [16][17][18][19][20][21] . Interestingly, the position dynamics were considered, which transformed the entire machine dynamics into a linear-time varying system that could be stabilized by an H 2 controller with a passive observer 22 .
Unlike the aforementioned approaches (designed in the a-b axis), the introduction of a rotational d-q transformation simplifies the controller design task considerably by removing of the nonlinearities of the model that rely on the motor position 23 . This method also enables to enlarge the operation range by controlling the negative d-axis current 24 . Moreover, several recent techniques for three-phase permanent machines, as in 25-30 ,

Hybrid-type stepping motor model
The stator of the stepping motors includes a-and b-phase, whose phase current and voltage are denoted as i x and v x , x = a, b , respectively. Applying the orthogonal coordinate transformation with the rotor position θ and each phase teeth number N r , respectively, it holds that 9,10,42 ∀t ≥ 0 , with the state variables : θ -rotor position (rad), ω -rotor speed (rad/s), and i x , x = d, q -current (A) and control variable : v x , x = d, q -voltage (V). The output torque T e (i q ) (Nm) is proportional to the q-axis current as T e (i q ) := K m i q with the torque coefficient K m . The load torque T L (Nm) acts as the mismatched disturbance depending on the operation conditions. The disturbances p x (i d , i q , ω) in modeling the back-EMF effect are defined as p d (i d , i q , ω) := LN r ωi q and p q (i d , i q , ω) := − (LN r i d + K m )ω with the stator inductance L (H). The remaining machine parameters are given by: J -inertia moment of the rotor (kgm 2 ), B -viscous friction (Nm/ rad/s), and R -stator resistance ( ).
To deal with the variations of parameter and load torque, nominal parameters denoted as (·) 0 are introduced for the speed and current dynamics (2)-(3) to be expressed as

Position-tracking control law
This study adopts the target position tracking behavior denoting θ * (different from the actual position measurement θ ) as the first-order system given by www.nature.com/scientificreports/ for the error θ * := θ ref − θ * , reference trajectory θ ref , and convergence rate ω pc > 0 (constant). The system (6) accomplishes the tracking objective; that is lim t→∞ θ * = θ ref , exponentially for any reference trajectory θ ref according to the convergence specification ω pc . Therefore, the tracking controller is designed to guarantee the control objective: lim t→∞ θ = θ * , exponentially, which is proved by analyzing the closed-loop dynamics in "Analysis" section.
Outer loop. Convergence rate boosting mechanism. The time -varying convergence ratio ω pc replaces its constant version ω pc in (6) as where ω pc (0) = ω pc , ω pc := ω pc −ω pc , and two design parameters γ pc > 0 and ρ pc > 0 determine the convergence rate booting and restoring natures, respectively. The time-varying nature of ω pc by the rule (8) makes the stability issue questionable, which is addressed in "Analysis" section with the boundedness property ω pc ≥ ω pc , ∀t ≥ 0. with the adjustable convergence rate pc > 0 . Note that the compensation term θ * is obtainable from the implementations of (7) and (8) such that θ * =ω pcθ * +θ ref . The proposed stabilizing solution (9) results in the controlled position dynamics: through the substitution of (9) to the open-loop error dynamics θ = −ω ref +ω +θ * , which is used in "Analysis" section to analyze the whole system stability and convergence properties considering all control dynamics in "Outer loop" and "Inner loop speed control" sections.
Inner loop speed control. This section presents the stabilizing solution for the second-order speed error dynamics given by with the coefficient c ω,0 := J 0 L 0 K m,0 (known) and lumped disturbance d ω : where w := ȧ and |w| ≤ w max , ∀t ≥ 0 , which corresponds to the genuine idea of this work to solve the model dependence problem in the observer design task. Defining the observer errors eω :=ω −ω obs and eã :=ã −ã obs with their estimates ω obs and ã obs , an acceleration error observer is proposed as with observer gains l obs,d > 0 (for disturbance attenuation level) and l obs,c > 0 (for estimation error convergence rate), whose pole-zero cancellation property results in the exponential convergence property lim t→∞ãobs =ã according to the first-order dynamics ėã = −l obs,c eã with a sufficient large l obs,d > 0 . See "Analysis" section for details specifying the admissible range for l obs,d . ∀t ≥ 0 , with design factor l > 0 . Figure 1 shows the controller structure.

Lemma 2
The convergence rate booster (7) and (8) achieves the lower bound on its initial value ω pc . i.e., Proof Consider the another form of (8) given by whose solution obtained by the both side integration above has a lower bound as ω pc = −γ pc ρ pcωpc + γ pc ρ pc ω pc + γ pc (θ * ) 2 , ∀t ≥ 0, Lemma 3 analyzes the observer error behavior used for the closed-loop convergence analysis.

Remark 1
The inequality V ε < 0 reveals that for some setting l obs,d > 0 . This leads to the reasoning process using the subsystem dynamics of (15) given by which concludes that verifying the exponential acceleration error estimation convergence used for the remaining analysis.
Lemma 4 analyzes the disturbance estimate behavior from DOB used for the closed-loop convergence analysis using the result of Lemma 3.

Lemma 4 The DOB (17) ensures that
Proof It follows from the output of DOB in the right side of (17) that where the DOB (17) yields the second equation and the last equation is obtained from the relationships (11) and (21), which completes the proof.
Closed-loop stability and convergence. Interestingly, the proposed controller results in the order reduction of the closed-loop speed dynamics by the stable pole-zero cancellation, which is proven in Lemma 5.

Lemma 5
The proposed voltage updating law of (16) allows the speed error to be satisfied: with the perturbations d ω,F and eã ,F such that Proof The controlled speed error system is obtained as with the combination of (11) and (16) and ω ref := 0 . Taking the Laplace transform, it holds that which yields that where the pole-zero cancellation from the factorizatioṅ  Eã(s) , which completes the proof.

Theorem 1 The proposed controller comprising
Proof The vector x := θωd ω,F eã ,F T leads to the augmented system given by The stability of A x picks an unique solution P x > 0 such that A T x P x + P x A x = −I , which defines the Lyapunov function candidate defined as The above augmented system and (22) gives its time derivative along the trajectories as with its upper bound by Young's inequality (e.g., p T q ≤ ε 2 �p� 2 + 1 2ε �q� 2 , ∀ε > 0): Remark 2 Based on the above analysis results, this remark finalizes this section by suggesting a tuning process of the proposed controller comprising the speed (inner) and position (outer) loops shown in Fig. 1 as follows: 1. (speed loop for steps 1-4) Using well-working speed controller, e.g., PI controller with a constant speed reference ω ref , tune the observer gains l obs,c and l obc,d for a rapid estimation error convergence in accordance with Remark 1; for example, first, choose l obs,c such that l obs,c ≥ 50 for ėω = −l obs,c eω and, second, increase l obs,d holding l obs,d ≫ l obs,c . 2. Tune the DOB gain l > 0 to assign the cut-off frequency ( l = 2πf l rad/s, equivalently, f l = l 2π Hz) for the transfer function D ω (s) D ω (s) = l s+l (obtained from (22) under the condition eã = 0 ); for example, choose f l ≥ 2 Hz such that l ≥ 2πf l = 12.6 rad/s. 3. Using the proposed speed controller (16) with a constant speed reference ω ref (for step 3 and 4), select f sc ∈ [10,30] yielding sc ∈ [2π10(= sc,min ), 2π30(= sc,max )] (e.g., sc = 2πf sc rad/s and f sc = sc 2π Hz); the maximum interval value may be increased depending on the hardware specification. 4. Increase the active damping coefficient k d (for example, k d ≥ 0.001 ) for an acceptable speed tracking response ω ≈ scω (some iteration between step 3 and 4 may be required). ; the maximum interval value may depend on the hardware specification. 6. Increase pc (for example, pc ≥ 10 ) until an acceptable position tracking response θ ≈ ω pcθ is obtained (some iteration between step 5 and 6 may be required). 7. Increase γ pc and ρ pc = κ pc γ pc with κ pc > 0 until the peak value and restoration rate of the convergence rate are acceptable; for example, choose γ pc ≥ 1 and κ pc ≥ (c ω,0 s 2 + (k d + c ω,0 sc )s + k d sc ) = (c ω,0 s + k d )(s + pc )

Experimental results
This section experimentally demonstrates the position-tracking performance improvements by comparison with an extant controller. A 10-W stepping motor embedding an encoder for position feedback (model:NK266E-02AT) and Texas Instrument (TI) LAUNCHXL-F28069M (digital signal processor) were used for experimental setup shown in Fig. 2 (see 41 for more detailed configuration). The controller tuning results are summarized as (convergence rate booster) f pc = 0.2 Hz such that ω pc (0) = ω pc = 2π0.2 rad/s, γ pc = 2 , ρ pc = 0.5/γ pc , (control gain) pc = 1.25 , sc = 125.6 , k d = 0.1 , (observer gain) l obs,c = 100 , l obs,d = 500 , and (DOB gain) l = 20 . This study chooses the FL controllers (in 30 ) for the comparison study, including the active damping and feed-forward term, given by: v x = K P,ccĩx + K I,cc    www.nature.com/scientificreports/ Tracking task. For the stair position reference, Fig. 3 demonstrates an improved tracking performance from the convergence rate boosting mechanism and performance recovery property proved in Theorem 1. Figure 4 presents the d-q axis current and observer error responses. Figure 5 shows the DOB and convergence rate booster responses. The intended convergence rate behavior improves the tracking performance as shown in Fig. 3.
Frequency response. The proposed controller robustly forces the position motion to be the first-order tracking error system (7) by the beneficial capability shown in Theorem 1. This section verifies this advantage for the sinusoidal reference signals 0.1, 0.2, and 0.3 Hz. Figure 6 shows that the proposed controller provides the desired position-tracking behavior without any magnitude and phase distortion unlike the FL controller.
Regulation task. To evaluate the regulation performance, a load torque of T L = 0.1 Nm was abruptly applied to the closed-loop system (by suddenly attaching the rotating wheel to the rotor) operating at θ ref ,Deg = 90 • under the three load conditions, such as light-, medium-, and heavy-sized fan. Figure 7 presents that the proposed technique accomplishes drastic regulation performance improvement under different loads compared with the FL controller that provides the magnified undershoots with oscillations and performance inconsistency for different load conditions. The corresponding q-axis current responses are compared in Fig. 8, which exhibits the improved current regulation performance by the proposed controller despite in the absence of current feedback.

Conclusions
The proposed current sensorless feedback system was driven by a PD-type controller incorporating the novel techniques, such as a convergence rate booster, angular acceleration error observer (model-free), and DOB without requiring the true motor parameters. This study has both proved the beneficial closed-loop properties and experimentally confirmed the practical advantages for tracking tasks. However, an acceptable setting for numerous design parameters should be identified through a systematic process, which is will be conducted in a future study.

Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.